the left member to be evaluated at \(w_k\) .
Here the second term in the left member is the well-known [1]}, [2]}
contribution from the Green function, where (referring to (REF ), (REF ))
\(\lbrace *dG^\omega \rbrace _{\rm renormalized}(z)=\Gamma _k*dR_{\rm robin}(z)+\sum _{j=1, j\ne k}^n\Gamma _j*dG(z,w_j),\) 
